3.388 \(\int \frac{A+B x^2}{x^{3/2} \left (a+b x^2\right )^3} \, dx\)

Optimal. Leaf size=322 \[ -\frac{5 (9 A b-a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{13/4} b^{3/4}}+\frac{5 (9 A b-a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{13/4} b^{3/4}}+\frac{5 (9 A b-a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{13/4} b^{3/4}}-\frac{5 (9 A b-a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{13/4} b^{3/4}}-\frac{5 (9 A b-a B)}{16 a^3 b \sqrt{x}}+\frac{9 A b-a B}{16 a^2 b \sqrt{x} \left (a+b x^2\right )}+\frac{A b-a B}{4 a b \sqrt{x} \left (a+b x^2\right )^2} \]

[Out]

(-5*(9*A*b - a*B))/(16*a^3*b*Sqrt[x]) + (A*b - a*B)/(4*a*b*Sqrt[x]*(a + b*x^2)^2
) + (9*A*b - a*B)/(16*a^2*b*Sqrt[x]*(a + b*x^2)) + (5*(9*A*b - a*B)*ArcTan[1 - (
Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(13/4)*b^(3/4)) - (5*(9*A*b - a
*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(13/4)*b^(3/4))
- (5*(9*A*b - a*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/
(64*Sqrt[2]*a^(13/4)*b^(3/4)) + (5*(9*A*b - a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b
^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*Sqrt[2]*a^(13/4)*b^(3/4))

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Rubi [A]  time = 0.521904, antiderivative size = 322, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454 \[ -\frac{5 (9 A b-a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{13/4} b^{3/4}}+\frac{5 (9 A b-a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{13/4} b^{3/4}}+\frac{5 (9 A b-a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{13/4} b^{3/4}}-\frac{5 (9 A b-a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{13/4} b^{3/4}}-\frac{5 (9 A b-a B)}{16 a^3 b \sqrt{x}}+\frac{9 A b-a B}{16 a^2 b \sqrt{x} \left (a+b x^2\right )}+\frac{A b-a B}{4 a b \sqrt{x} \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^2)/(x^(3/2)*(a + b*x^2)^3),x]

[Out]

(-5*(9*A*b - a*B))/(16*a^3*b*Sqrt[x]) + (A*b - a*B)/(4*a*b*Sqrt[x]*(a + b*x^2)^2
) + (9*A*b - a*B)/(16*a^2*b*Sqrt[x]*(a + b*x^2)) + (5*(9*A*b - a*B)*ArcTan[1 - (
Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(13/4)*b^(3/4)) - (5*(9*A*b - a
*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(13/4)*b^(3/4))
- (5*(9*A*b - a*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/
(64*Sqrt[2]*a^(13/4)*b^(3/4)) + (5*(9*A*b - a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b
^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*Sqrt[2]*a^(13/4)*b^(3/4))

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Rubi in Sympy [A]  time = 90.2279, size = 298, normalized size = 0.93 \[ \frac{A b - B a}{4 a b \sqrt{x} \left (a + b x^{2}\right )^{2}} + \frac{9 A b - B a}{16 a^{2} b \sqrt{x} \left (a + b x^{2}\right )} - \frac{5 \left (9 A b - B a\right )}{16 a^{3} b \sqrt{x}} - \frac{5 \sqrt{2} \left (9 A b - B a\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{128 a^{\frac{13}{4}} b^{\frac{3}{4}}} + \frac{5 \sqrt{2} \left (9 A b - B a\right ) \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{128 a^{\frac{13}{4}} b^{\frac{3}{4}}} + \frac{5 \sqrt{2} \left (9 A b - B a\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{13}{4}} b^{\frac{3}{4}}} - \frac{5 \sqrt{2} \left (9 A b - B a\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{13}{4}} b^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)/x**(3/2)/(b*x**2+a)**3,x)

[Out]

(A*b - B*a)/(4*a*b*sqrt(x)*(a + b*x**2)**2) + (9*A*b - B*a)/(16*a**2*b*sqrt(x)*(
a + b*x**2)) - 5*(9*A*b - B*a)/(16*a**3*b*sqrt(x)) - 5*sqrt(2)*(9*A*b - B*a)*log
(-sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqrt(a) + sqrt(b)*x)/(128*a**(13/4)*b**(3/
4)) + 5*sqrt(2)*(9*A*b - B*a)*log(sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqrt(a) +
sqrt(b)*x)/(128*a**(13/4)*b**(3/4)) + 5*sqrt(2)*(9*A*b - B*a)*atan(1 - sqrt(2)*b
**(1/4)*sqrt(x)/a**(1/4))/(64*a**(13/4)*b**(3/4)) - 5*sqrt(2)*(9*A*b - B*a)*atan
(1 + sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(64*a**(13/4)*b**(3/4))

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Mathematica [A]  time = 0.495145, size = 285, normalized size = 0.89 \[ \frac{\frac{32 a^{5/4} x^{3/2} (a B-A b)}{\left (a+b x^2\right )^2}+\frac{5 \sqrt{2} (a B-9 A b) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{b^{3/4}}+\frac{5 \sqrt{2} (9 A b-a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{b^{3/4}}+\frac{10 \sqrt{2} (9 A b-a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{b^{3/4}}-\frac{10 \sqrt{2} (9 A b-a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{b^{3/4}}+\frac{8 \sqrt [4]{a} x^{3/2} (5 a B-13 A b)}{a+b x^2}-\frac{256 \sqrt [4]{a} A}{\sqrt{x}}}{128 a^{13/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x^2)/(x^(3/2)*(a + b*x^2)^3),x]

[Out]

((-256*a^(1/4)*A)/Sqrt[x] + (32*a^(5/4)*(-(A*b) + a*B)*x^(3/2))/(a + b*x^2)^2 +
(8*a^(1/4)*(-13*A*b + 5*a*B)*x^(3/2))/(a + b*x^2) + (10*Sqrt[2]*(9*A*b - a*B)*Ar
cTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/b^(3/4) - (10*Sqrt[2]*(9*A*b - a*B)
*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/b^(3/4) + (5*Sqrt[2]*(-9*A*b + a
*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/b^(3/4) + (5*Sqr
t[2]*(9*A*b - a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/b
^(3/4))/(128*a^(13/4))

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Maple [A]  time = 0.026, size = 363, normalized size = 1.1 \[ -2\,{\frac{A}{{a}^{3}\sqrt{x}}}-{\frac{13\,{b}^{2}A}{16\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{2}}{x}^{{\frac{7}{2}}}}+{\frac{5\,Bb}{16\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{2}}{x}^{{\frac{7}{2}}}}-{\frac{17\,Ab}{16\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{2}}{x}^{{\frac{3}{2}}}}+{\frac{9\,B}{16\,a \left ( b{x}^{2}+a \right ) ^{2}}{x}^{{\frac{3}{2}}}}-{\frac{45\,\sqrt{2}A}{128\,{a}^{3}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{45\,\sqrt{2}A}{64\,{a}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{45\,\sqrt{2}A}{64\,{a}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{5\,\sqrt{2}B}{128\,{a}^{2}b}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{5\,\sqrt{2}B}{64\,{a}^{2}b}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{5\,\sqrt{2}B}{64\,{a}^{2}b}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)/x^(3/2)/(b*x^2+a)^3,x)

[Out]

-2*A/a^3/x^(1/2)-13/16/a^3/(b*x^2+a)^2*x^(7/2)*b^2*A+5/16/a^2/(b*x^2+a)^2*x^(7/2
)*b*B-17/16/a^2/(b*x^2+a)^2*A*x^(3/2)*b+9/16/a/(b*x^2+a)^2*B*x^(3/2)-45/128/a^3/
(a/b)^(1/4)*2^(1/2)*A*ln((x-(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2))/(x+(a/b)^(1
/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2)))-45/64/a^3/(a/b)^(1/4)*2^(1/2)*A*arctan(2^(1/2)
/(a/b)^(1/4)*x^(1/2)+1)-45/64/a^3/(a/b)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(a/b)^(1/
4)*x^(1/2)-1)+5/128/a^2/b/(a/b)^(1/4)*2^(1/2)*B*ln((x-(a/b)^(1/4)*x^(1/2)*2^(1/2
)+(a/b)^(1/2))/(x+(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2)))+5/64/a^2/b/(a/b)^(1/
4)*2^(1/2)*B*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+5/64/a^2/b/(a/b)^(1/4)*2^(1/2
)*B*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/((b*x^2 + a)^3*x^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.258484, size = 1177, normalized size = 3.66 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/((b*x^2 + a)^3*x^(3/2)),x, algorithm="fricas")

[Out]

1/64*(20*(B*a*b - 9*A*b^2)*x^4 - 128*A*a^2 + 36*(B*a^2 - 9*A*a*b)*x^2 - 20*(a^3*
b^2*x^4 + 2*a^4*b*x^2 + a^5)*sqrt(x)*(-(B^4*a^4 - 36*A*B^3*a^3*b + 486*A^2*B^2*a
^2*b^2 - 2916*A^3*B*a*b^3 + 6561*A^4*b^4)/(a^13*b^3))^(1/4)*arctan(-a^10*b^2*(-(
B^4*a^4 - 36*A*B^3*a^3*b + 486*A^2*B^2*a^2*b^2 - 2916*A^3*B*a*b^3 + 6561*A^4*b^4
)/(a^13*b^3))^(3/4)/((B^3*a^3 - 27*A*B^2*a^2*b + 243*A^2*B*a*b^2 - 729*A^3*b^3)*
sqrt(x) - sqrt((B^6*a^6 - 54*A*B^5*a^5*b + 1215*A^2*B^4*a^4*b^2 - 14580*A^3*B^3*
a^3*b^3 + 98415*A^4*B^2*a^2*b^4 - 354294*A^5*B*a*b^5 + 531441*A^6*b^6)*x - (B^4*
a^11*b - 36*A*B^3*a^10*b^2 + 486*A^2*B^2*a^9*b^3 - 2916*A^3*B*a^8*b^4 + 6561*A^4
*a^7*b^5)*sqrt(-(B^4*a^4 - 36*A*B^3*a^3*b + 486*A^2*B^2*a^2*b^2 - 2916*A^3*B*a*b
^3 + 6561*A^4*b^4)/(a^13*b^3))))) - 5*(a^3*b^2*x^4 + 2*a^4*b*x^2 + a^5)*sqrt(x)*
(-(B^4*a^4 - 36*A*B^3*a^3*b + 486*A^2*B^2*a^2*b^2 - 2916*A^3*B*a*b^3 + 6561*A^4*
b^4)/(a^13*b^3))^(1/4)*log(125*a^10*b^2*(-(B^4*a^4 - 36*A*B^3*a^3*b + 486*A^2*B^
2*a^2*b^2 - 2916*A^3*B*a*b^3 + 6561*A^4*b^4)/(a^13*b^3))^(3/4) - 125*(B^3*a^3 -
27*A*B^2*a^2*b + 243*A^2*B*a*b^2 - 729*A^3*b^3)*sqrt(x)) + 5*(a^3*b^2*x^4 + 2*a^
4*b*x^2 + a^5)*sqrt(x)*(-(B^4*a^4 - 36*A*B^3*a^3*b + 486*A^2*B^2*a^2*b^2 - 2916*
A^3*B*a*b^3 + 6561*A^4*b^4)/(a^13*b^3))^(1/4)*log(-125*a^10*b^2*(-(B^4*a^4 - 36*
A*B^3*a^3*b + 486*A^2*B^2*a^2*b^2 - 2916*A^3*B*a*b^3 + 6561*A^4*b^4)/(a^13*b^3))
^(3/4) - 125*(B^3*a^3 - 27*A*B^2*a^2*b + 243*A^2*B*a*b^2 - 729*A^3*b^3)*sqrt(x))
)/((a^3*b^2*x^4 + 2*a^4*b*x^2 + a^5)*sqrt(x))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**2+A)/x**(3/2)/(b*x**2+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.272413, size = 405, normalized size = 1.26 \[ -\frac{2 \, A}{a^{3} \sqrt{x}} + \frac{5 \, B a b x^{\frac{7}{2}} - 13 \, A b^{2} x^{\frac{7}{2}} + 9 \, B a^{2} x^{\frac{3}{2}} - 17 \, A a b x^{\frac{3}{2}}}{16 \,{\left (b x^{2} + a\right )}^{2} a^{3}} + \frac{5 \, \sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} B a - 9 \, \left (a b^{3}\right )^{\frac{3}{4}} A b\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{64 \, a^{4} b^{3}} + \frac{5 \, \sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} B a - 9 \, \left (a b^{3}\right )^{\frac{3}{4}} A b\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{64 \, a^{4} b^{3}} - \frac{5 \, \sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} B a - 9 \, \left (a b^{3}\right )^{\frac{3}{4}} A b\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{128 \, a^{4} b^{3}} + \frac{5 \, \sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} B a - 9 \, \left (a b^{3}\right )^{\frac{3}{4}} A b\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{128 \, a^{4} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/((b*x^2 + a)^3*x^(3/2)),x, algorithm="giac")

[Out]

-2*A/(a^3*sqrt(x)) + 1/16*(5*B*a*b*x^(7/2) - 13*A*b^2*x^(7/2) + 9*B*a^2*x^(3/2)
- 17*A*a*b*x^(3/2))/((b*x^2 + a)^2*a^3) + 5/64*sqrt(2)*((a*b^3)^(3/4)*B*a - 9*(a
*b^3)^(3/4)*A*b)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4
))/(a^4*b^3) + 5/64*sqrt(2)*((a*b^3)^(3/4)*B*a - 9*(a*b^3)^(3/4)*A*b)*arctan(-1/
2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(a^4*b^3) - 5/128*sqrt(
2)*((a*b^3)^(3/4)*B*a - 9*(a*b^3)^(3/4)*A*b)*ln(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x
+ sqrt(a/b))/(a^4*b^3) + 5/128*sqrt(2)*((a*b^3)^(3/4)*B*a - 9*(a*b^3)^(3/4)*A*b)
*ln(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^4*b^3)